Asymptotics of some quantum invariants of the Whitehead chains (1912.10638v2)

Abstract: In this paper, we study the asymptotics of the colored Jones polynomials of the Whitehead chains with one belt colored by $M_1$ and all the clasps colored by $M_2$ evaluated at the $(N+1/2)$-th root of unity $t=e^{\frac{2\pi i}{N+1/2}}$, where $M_1$ and $M_2$ are sequences of integers in $N$. By considering the limiting ratios, $s_1$ and $s_2$, of $M_1$ and $M_2$ to $(N+1/2)$, we show that the exponential growth rate of the invariants coincides with the hyperbolic volume of the link complement equipped with certain (possibly incomplete) hyperbolic structure parametrized by $s_1$ and $s_2$. In the proof we figure out the correspondence between the critical point equations of the potential functions and the hyperbolic gluing equations of certain triangulations of the link complements. Furthermore, we discover a connection between the potential function, the theory of angle structures and the covolume function. As a corollary, we prove the volume conjecture for the Turaev-Viro invariants for all Whitehead chains complements.

• The paper confirms the volume conjecture by linking the asymptotics of colored Jones polynomials to the hyperbolic volume of Whitehead chain complements.
• Saddle point and potential function analyses yield rigorous asymptotic expansion formulas that connect quantum invariants with classical hyperbolic geometry.
• The study extends its framework to a broader class of Whitehead chains, paving the way for further research in quantum topology and knot theory.

Asymptotics of Some Quantum Invariants of the Whitehead Chains: An Overview

The paper "Asymptotics of some quantum invariants of the Whitehead chains" by Ka Ho Wong explores the asymptotic behaviors of quantum invariants specifically related to the family of topological objects known as Whitehead chains. The paper navigates the interrelations between the colored Jones polynomials evaluated at certain roots of unity and the hyperbolic structures of link complements. Here, the concepts from hyperbolic geometry, quantum topology, and knot theory intersect, leading to profound insights into the volume conjecture, a prominent unifying theme across these domains.

Volume Conjecture and Quantum Invariants

The volume conjecture, originally posited by Kashaev and later generalized, proposes that certain quantum invariants of link complements, such as the colored Jones polynomials, relate directly to the volumes of hyperbolic 3-manifolds. Wong's paper confirms the conjecture for a specific class of links, namely the Whitehead chains, by examining their colored Jones polynomials evaluated at $(N+1/2)$-th roots of unity $t = e^{\frac{2\pi i}{N+1/2}}$. A key result is that the exponential growth rate of these invariants aligns with the hyperbolic volume of the link complement, parameterized by sequences $M_1$ and $M_2$, which define the colors assigned to link components.

Analytical Approach

To achieve these results, Wong employs critical point and saddle point methods, seamlessly integrating them with the potential functions framework. For these, explicit correspondences are drawn between the critical point equations derived from the potential functions and the hyperbolic gluing equations for certain triangulations of link complements. This not only aids in the proof of the volume conjecture specific to Whitehead chains but also extends the understanding of the intricate links between topological, geometric, and quantum properties of knots.

Results and Extensions

The paper provides rigorous asymptotic expansion formulas for these quantum invariants. For example, it demonstrates that the colored Jones polynomials of such links exhibit critical values identical to the hyperbolic volume of the link complement. Additionally, Wong explores connections between the potential and the Lobachevsky functions, providing new insights into the geometric interpretations of quantum knots.

Further, the paper extends the analysis to a broader class of links labeled as $W_{a,1,c,d}$ where $a, c, d$ are integers dictating the specific makeup of Whitehead chains. The significant result here is the asymptotic alignment of the volume conjecture for Turaev-Viro invariants with non-hyperbolic link complements, a challenging regime generally in these asymptotic studies.

Implications and Future Directions

The results of this paper have important implications for theoretical and quantum topology, suggesting possible generalizations of the volume conjecture to broader configurations and relations between distinct link invariants. Moreover, it opens up avenues for first-principles proofs of quantum knot invariants mapping to classical geometry. The differential formula exhibited for the potential function could further bridge connections to covolume functions and angle structures, potentially illuminating new avenues in hyperbolic geometry and topological quantum field theories.

Wong's paper stands as a vibrant piece of research interweaving multiple advanced concepts, forming a robust foundation for further explorations in the asymptotic behaviors of quantum invariants and their links to hyperbolic 3-manifolds. The techniques developed and results obtained provide significant groundwork for the interpretation and expansion of the volume conjecture in knot theory.